Area of plane surfaces - Formulas

Area of plane surfaces - Important Formulas, shortcuts and Tricks:

1. Rectangle:

A rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle.

(A) Area of a rectangle = (Length × Breadth)

∴Length = (Area/Breadth) and Breadth = (Area/Length).

(B) Perimeter of a rectangle = 2(Length + Breadth)

(C) Diagonal of a rectangle = √(Length² + Breadth²) = √(l² + b²)

(D) Area of 4 walls of a room = 2 (Length + Breadth) × Height

2. Square:

Asquare is any quadrilateral with four right angles and all of its sides are of equal length.

(A) Diagonal of a square = √2 × a

(B) Area of a square = (side)² = ½ (diagonal)²

(C) Perimeter of a square = 4a, where a is each side

2. Triangle:

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted by ΔABC

(A) Area of a triangle = ½ × Base × Height

(B) Area of a triangle =

Where a, b, c are the sides of triangle and s = ½ (a + b + c)

(C) Area of an equilateral triangle = √3/4× (side)²

(D) Height of a equilateral triangle = √3/2× a

(E) Radius of incircle of an equilateral triangle of side a = a/2√3

(F) Radius of circumcircle of an equilateral triangle of side a = a/√3

(G) Radius of in-circle of a triangle of area Δ and semi-perimeter s = Δ/s

3. Parallelogram:

A parallelogram is a (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

(A) Area of Parallelogram = (Base × Height)

(B) Perimeter of Parallelogram = 2 × (sum of adjacent sides)

4. Rhombus:

If all the sides of a Parallelogram are equal, it is called a rhombus. The diagonals of rhombus bisect each other at right angles.

If d₁ and d₂ are the diagonals of a rhombus, then

(A) Side of rhombus = ½ × √(d₁² + d₂²)

(B) Area of rhombus = ½ × (product of diagonals) = ½ × d₁d₂

(C) Perimeter of rhombus = 2 × √(d₁² + d₂²)

4. Trapezium:

A trapezium is a quadrilateral that has only one pair of parallel sides.

(A) Area of a trapezium = ½ × (sum of parallel sides) × distance between them.

5. Circle:

A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point always remains same. The fixed point is called the centre and the constant distance is known as the radius of the circle. If r = radius, then,

(A) Area of a circle = πr², where, r is the radius.

(B) Circumference of a circle = 2πr

(C) Length of an arc = 2πrθ/360, where θ is the central angle.

(D) Area of a sector = ½(arc × r) = πr²θ/360

(E) Circumference of a semicircle = πr

(F) Area of a semicircle = πr²/2

(G) Area of a quadrant circle = πr²/4

(H) If two circles touch internally/externally, then the distance between their centres is equal to difference/sum

(I) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.

(J) The number of revolutions completed by rotating a wheel in one minute = Distance moved in one minute/ Circumference